## Taming Non-Compositionality Using New Binders

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Citations: | 2 - 0 self |

### BibTeX

@MISC{Prost_tamingnon-compositionality,

author = {F. Prost},

title = {Taming Non-Compositionality Using New Binders},

year = {}

}

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### Abstract

Abstract. We propose an extension of the traditional λ-calculus in which terms are used to control an outside computing device (quantum computer, DNA computer...). We introduce two new binders: ν and ρ. In νx.M, x denotes an abstract resource of the outside computing device, whereas in ρx.M, x denotes a concrete resource. These two binders have different properties (in terms of α-conversion, scope extrusion, convertibility) than the ones of standard λ-binder. We illustrate the potential benefits of our approach with a study of a quantum computing language in which these new binders prove meaningful. We introduce a typing system for this quantum computing framework in which linearity is only required for concrete quantum bits offering a greater expressiveness than previous propositions. 1

### Citations

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Quantum computation and quantum information
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Citation Context ... it requires sophisticated approach very different from usual typing systems (see [11]). Things can get even more complicated when the global state is not classical. If you consider quantum computing =-=[17]-=-, the global state is an array of quantum bits. Several functional programming language have been proposed for quantum computation (e.g. [22, 21, 6]). In a quantum functional programing language, quan... |

1184 |
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- 1984
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Citation Context ...yping system for this quantum computing framework in which linearity is only required for concrete quantum bits offering a greater expressiveness than previous propositions. 1 Introduction λ-calculus =-=[8]-=- is a particularly well suited formalism to study functional programs. Any λ-variable can be replaced by any λ-term in the pure λ-calculus. λ-abstraction can be seen in the following way: the term λx.... |

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(Show Context)
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Citation Context ...ing” of x. This inner compositionality of λ-calculus is reflected by the success of type systems for λ-caculi [9]. Those type systems can be used with good effect to statically analyze programs (e.g. =-=[2, 20, 19]-=-). Some λ-calculus variants have been developed to study functional programing languages including imperative features like ML (e.g. [3, 14, 18]). The idea is to introduce a global state and to refer ... |

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54 | A lambda calculus for quantum computation - Tonder |

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Citation Context ... programing system based on λGS. We suppose that the reader has basic knowledges of quantum computing (see [17]). It is a functional programming language for quantum computers based on the QRAM model =-=[13]-=-. The physical device of λ Q GS is an array of qbits. n qbits are represented as normalized vector of the Hilbert space ⊗n i=1C2 noted as a ket vector : |ϕ〉. quantum actions are standard unitary quant... |

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Citation Context ... state is not classical. If you consider quantum computing [17], the global state is an array of quantum bits. Several functional programming language have been proposed for quantum computation (e.g. =-=[22, 21, 6]-=-). In a quantum functional programing language, quantum bits can be entangled by side effects that go beyond the lexical scope of their definitions. and thus such programing languages exhibit a highly... |

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Citation Context ... state is not classical. If you consider quantum computing [17], the global state is an array of quantum bits. Several functional programming language have been proposed for quantum computation (e.g. =-=[22, 21, 6]-=-). In a quantum functional programing language, quantum bits can be entangled by side effects that go beyond the lexical scope of their definitions. and thus such programing languages exhibit a highly... |

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Citation Context ...d with good effect to statically analyze programs (e.g. [2, 20, 19]). Some λ-calculus variants have been developed to study functional programing languages including imperative features like ML (e.g. =-=[3, 14, 18]-=-). The idea is to introduce a global state and to refer to it via variable names. Name management is handled via standard λ-abstraction mechanism. Typically in ML-like languages [16] one can define a ... |

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Citation Context ...t is syntacticly different from x) then (Q z) increments x by 2. Static analysis of such programs is not an easy task, it requires sophisticated approach very different from usual typing systems (see =-=[11]-=-). Things can get even more complicated when the global state is not classical. If you consider quantum computing [17], the global state is an array of quantum bits. Several functional programming lan... |

12 |
A new calculus of contexts
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- 2005
(Show Context)
Citation Context ...ful since it allows the definition of a type system for a quantum calculus that is more flexible than previous propositions of the literature [21, 6]. This work compares to the NEW calculus of Gabbay =-=[12]-=- where a context substitution is introduced through a new binder. Nevertheless, this works principally addresses problems related to α-equivalence and does not talk about scope extrusion. Another rela... |

5 |
A static calculus of dependencies for the λ-cube
- Prost
- 2000
(Show Context)
Citation Context ...ing” of x. This inner compositionality of λ-calculus is reflected by the success of type systems for λ-caculi [9]. Those type systems can be used with good effect to statically analyze programs (e.g. =-=[2, 20, 19]-=-). Some λ-calculus variants have been developed to study functional programing languages including imperative features like ML (e.g. [3, 14, 18]). The idea is to introduce a global state and to refer ... |

3 |
Quantum Computation Explained to my Mother
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Citation Context ...r programming flexibilitysthan previous propositions of the literature. Finally, we conclude and discuss future works in section 4. Knowledge of basics of quantum computation is supposed. We refer to =-=[7, 1]-=- for quantum computing tutorials. 2 Global State Calculus 2.1 λGS ideas A λGS calculus is defined relatively to an outside computing device on which computations/actions can be performed (such as a qu... |

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1 |
The qν and qνK calculi : name capture and control
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- 2003
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Citation Context ...rough a new binder. Nevertheless, this works principally addresses problems related to α-equivalence and does not talk about scope extrusion. Another related work is the short note of Baro and Maurel =-=[10]-=- in which the λ binder is split into two constructions: a pure binder, ν and a combinator for the implementation of the β-reduction. In this case too the scope extrusion is not addressed, and there is... |